Question 33579
This is an example of a "geometric sequence" in which the common ratio, r = 2

1, 2, 4, 8, 16, 32, 64, ... {{{r^(n-1)}}} where n = the number of the square.

So, for the 32nd square, Mr Brown would have to put {{{2^(32-1)}}} pennies.
This would amount to {{{2^31 = 2147483648}}}pennies.
Divide this by 100 to get the number of dollars.
a) $21,474,836.48

b) To find the sum of the pennies on 32 squares, you need to find the partial sum of the of the first 32 terms of the geometric sequence, called a geometric series. The partial sum of the first n terms of a geometric series is given by{{{Sn = a1(1-r^n)/(1-r)}}} where: a1 is the first term (1), r is the common ratio (2) and, for 32 terms (squares), n = 32.

So, with a1 = 1 and r = 2, this simplifies to{{{Sn = r^n - 1}}}
If the checker board had only 32 squares, it would require{{{2^32 - 1 = 4294967296 - 1}}} pennies to fill the board. This would amount to:
$42,949,672.95

To fill the entire board of 64 squares would take{{{2^64 - 1}}}pennies.
{{{2^64 - 1 = 18446744073709551616 - 1}}} pennies. This would amount to:
$184,467,440,737,095,516.15